inner stability final
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Good paper on inner stabilityTRANSCRIPT
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Stability and error analysis of the anti-causalrealization of periodic inverse systems
Xiaofang ChenSchool of Electrical andElectronic Engineering
Nanyang Technological UniversityNanyang Avenue, Singapore 639798
Email: [email protected]
Cishen ZhangSchool of Electrical andElectronic Engineering
Nanyang Technological UniversityNanyang Avenue, Singapore 639798
Email: [email protected]
Jingxin ZhangDepartment of Electrical and
Computer Systems EngineeringMonash University,VIC 3800, Australia
Abstract A causal realization of an inverse system can beunstable and an anti-casual realization is to deal with thisproblem to provide a numerically stable procedure to inversethe system and compute its input signal. In this paper, weconsider the anti-causal realization of the inverse of discrete timelinear periodic systems obtained by an outer-inner factorizationapproach. It is shown that the outer-inner factorization can resultin a stable anti-causal realization. It also derives a formula ofthe inversion error, which can show that the inversion error isinevitable due to the anti-causal reversal operation.
I. INTRODUCTION
Inverse systems have important applications in signal pro-cessing, control, filtering and data coding, which may alsobe called equalization or deconvolution systems in differentcircumstance. The problem of inverting a linear periodic time-varying (LPTV) system has attracted considerable attentions,see for example, [1]-[5]. The anti-causal realization of theinverse system is to deal with the stability problem in theinverse system, which was earlier studied and applied tomultirate filter banks and its implementation [6], [7]. In theseworks, a state space representation is employed to induce theanti-causal inverse.
The inner-outer factorization is a useful technique to dealwith the system inverse problem. In [8], a method is proposedto construct optimal causal approximate inverse for discretetime single-input single-output (SISO) causal periodic filtersin the presence of measurement noise. The method is basedon an inner-outer factorization of the plant rational matrix.In [9]-[11], van der Veen and Dewilde present an innerouter factorization method for linear discrete time-varyingsystems. Instead of factorizing the transfer function matrix,this method factorizes the system time domain operator. It hasan computational advantage that all computations are done byusing the state space realization.
While [9] presents the concept and theory necessary for thedevelopment of the inner-outer factorization method for lineartime varying systems, it does not provide explicit stabilityanalysis of the factorized systems.
In this paper, we will analyze the stability of the anti-causalrealization of the periodic inverse system which is obtainedfrom the outer-inner factorization of the inverse system in
the state space realization. A periodic Lyapunov equation isintroduced to the stability analysis. We will further examinethe error introduced by using the anti-causal realization of thesystem inverse to compute the system input signal. Such anerror analysis has not been presented in the existing literature.
The rest of the paper is organized as follows. Section IIpresents necessary preliminaries. The anti-causal realizationof the inverse system is presented in III. Section IV analyzesthe stability of the inner factor and its anti-causal inverse. Andthe system inversion error is analyzed in section V. SectionVI illustrates the system error by examples before conclusionin section VII.
II. PRELIMINARIES
The following notations are used in this paper. I denotesthe identity matrix of an appropriate dimension, denotesthe hermitian conjugate, Rmn denotes the real space ofdimension (m n). The factorization of an operator T intoT = ToV is called an outer-inner factorization, where To isthe left outer factor and V is an isometry satisfying V V = I .The matrix V is inner if and only if T is full column rank.
Let u and y be the input and output of the system T suchthat y = uT . The idealized inverse problem is to find theinverse system T1 of T , such that the input u is computedand obtained from the measurement of y, i.e. u = yT1.This idealized case is often practically not feasible since thereis no guarantee that T1 is a stable system. The anti-causalrealization of the inverse is a method for stably computing theinput u and its restriction is that u should be of finite length.
The state realization of a SISO LPTV causal system T iswritten as:
xk+1 = xkAk + ukBk,yk = xkCk + ukDk,
(1)
where uk R11 is the input and yk R11 is the output. Ifthe state row vector xk has a time varying dimension R1bk ,then the dimensions of Ak, Bk, Ck and Dk are respectivelyRbkbk+1 , R1bk+1 , Rbk1 and R11. In this paper, we con-sider the constant state dimension R1n. For the N-periodictime varying system, the state matrices satisfy Ak+N = Ak,Bk+N = Bk, Ck+N = Ck, and Dk+N = Dk. It follows from
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the state equation (1) that entries of the system operator T aregiven by
Ti,j ={
Di, i = j,BiAi+1 Aj1Cj , i < j. (2)
The state realization of an N-periodic LPTV anti-causalsystem T
is written as:
xk1 = x
kA
k + u
kB
k,
yk = x
kC
k + u
kD
k.
(3)
The dimensions of all the matrices in the realization must becompatible. If the input u
k R11, the output y
k R11
and the state row vector xk has a time varying dimension
R1bk , the dimensions of Ak, Bk, Ck and Dk are respectivelyRb
kb
k1 , R1b
k+1 , Rbk1 and R11. The state matrices are
also in period of N and entries of the system operator T
aregiven by
Ti,j =
{Di, i = j,
BiA
i1 A
j+1C
j , i > j.
(4)
Definition 1: The causal system T with the state equa-tion (1) is stable if all the eigenvalues of the lifted matrixA = Ak Ak+N1 are inside the unit disk, or the greatesteigenvalue of A is small than unity, i.e.
lA = maxi
i(A) < 1,
where i(A) for i = 0, 1, , N 1 are the eigenvalues ofthe lifted matrix A.
The stability of the anti-causal system T
with state matrices{Ak, B
k, C
k, D
k} is defined as follows:
Definition 2: The anti-causal realization system T
is stableif
lA = maxii(A
) < 1,
where A= A
k+N1 A
k is the lifted matrix, and i(A
)
for i = 0, 1, , N 1 are the eigenvalues of A .In this paper, we consider the stable causal-anticausal re-
alization of the inverse of the causal SISO LPTV system Twith the state equation (1) and entries represented in (2). Itis assume that T is a stable system. In general and followingfrom the method in [9], the system T can be factorized intoa left outer factor To and the inner factor V , such that
T = ToV. (5)
The above outer-inner factorization is computed by the back-ward recursive LQ factorization[
Dk BkYk+1Ck AkYk+1
]=[Do,k 0 0Co,k Yk 0
]Wk, (6)
where the initial starting point Yk+1 is obtained by the theperiodic Riccati equation Mk+1 = Yk+1Y k+1, which is writtenas
Mk = AkMk+1Ak + CkCk
[AkMk+1Bk + CkDk](DkDk +BkMk+1Bk)[DkCk +BkMk+1Ak].
The numerical solution of the periodic Riccati equation is pre-sented in [13]. The partitioning of the outer-inner factorizationis such that Do,k and Yk have full column rank, and Wk isunitary matrix with partitioning in the form
Wk =
Dv,k Bv,kCv,k Av,kd d
,where d can be any matrix.
Since Wk is a unitary matrix, we can obtain the followingequations.
Av,kAv,k + Cv,kC
v,k = I, Av,kB
v,k + Cv,kD
v,k = 0,
Bv,kAv,k +Dv,kC
v,k = 0, Bv,kB
v,k +Dv,kD
v,k = I.
(7)The state-space realization of the outer factor To is given
by {Ak,Bk,Co,k,Do,k}. And the realization of the inner factorV is given by {Av,k, Bv,k, Cv,k, Dv,k}. The outer-inner fac-torization as given in the above will be the basis for the stableanti-causal realization of the inverse system of T .
III. ANTI-CAUSAL REALIZATION OF THE INVERSE SYSTEMWe now derive the stable anti-causal realization of the
inverse system of the causal stable LPTV system T . It is notedthat the anti-casual realization can only permit practically finitestep of computation and we assume that the number of theinput data in u is L + 1. Thus the input sequence of u is acollection of uk for 0 k L, as follows.
u =[u0 u1 u2 uL
].
Mathematically, the inverse of the transfer operator T is acombination of the inverse of the inner factor and the inverseof the outer factor, resulting in
u = yV 1T1o ,= uToV V 1T1o .
However, the direct casual inverse V 1 of the inner factor Vis an unstable system and practically unfeasible, which will beshown in the next section. In real computation, the unstablecausal factor of the inverse system can be realized anti-causallyto deal with the computational divergence problem. A resulton the anti-causal realization is stated in the following.
Theorem 1: The anti-causal realization of the inverse ofthe inner factor is V = V , where V is anti-causal andis the Hermitian conjugates V with a state space realization{Av,k, Cv,k, Bv,k, Dv,k}.
Proof: The casual realization of the N-periodic inner factorcan be rewritten into an operator form by assembling the statematrices {Av,k, Bv,k, Cv,k, Dv,k} as diagonal operators:
Av =
. . . 0
Av,k
0. . .
Cv =
. . . 0Cv,k
0. . .
Bv =
. . . 0
Bv,k
0. . .
Dv =
. . . 0Dv,k
0. . .
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Hence, the transfer function of the inner factor V is given as
V (z) = Dv +Bvz(I Avz)1Cv,and the transfer function of the anti-causal realization of theinverse of the inner factor is shown as
V (z) = Dv + Cv (I z)Av)1zBv .
In view of (7), we can obtain DvDv = I BvBv , CvDv =AvBv , DvCv = BvAv , and AvAv + CvCv = I . Usingthese, we have
V (z)V (z)= [Dv +Bvz(I Avz)1Cv][Dv + Cv (I zAv)1zBv ]= DvDv +DvCv (I zAv)1zBv+Bvz(I Avz)1CvDv+Bvz(I Avz)1CvCv (I zAv)1zBv= DvDv Bvz(I Avz)1AvBvBvAv(I zAv)1zBv+Bvz(I Avz)1CvCv (I zAv)1zBv= DvDv +Bvz(I Avz)1[Avz(I zAv)(I Avz)zAv + CvCv ](I zAv)1zBv= DvDv +Bvz(I Avz)1(I Avz)(I zAv)(I zAv)1zBv= I
Hence, V with the state matrices {Av,k, Cv,k, Bv,k, Dv,k} isthe anti-causal inverse of V . 2
In the rest of this paper, we use the notations V and Toas the anti-casual inverse of the inner factor and the causalinverse of the outer factor, respectively.
With u being the input to the LPTV system T and with theanti-casual realization of the inverse system, the output u theinverse system is written as
u = uToV V To
We define the error of the inverse system as follows.Definition 3: The system error is defined as the difference
of the the reconstructed signal (output signal from the inversesystem) and the input signal, that is,
e = u u.IV. STABILITY ANALYSIS OF THE INNER FACTOR
The stability of an LPTV system can be examined by usingthe periodic Lyapunov equation technique presented in [12].The following lemma is deduced from [12] for the stabilityanalysis of the LPTV inverse system.
Lemma 1: The N-periodic LPTV system T in the form (1)is stable if and only if there exists an N-periodic system matrixQk Rnn with Qk = Qk 0 and Qk+N = Qk for k =0, 1, , N 1 such that the following periodic Lyapunovequation is satisfied:
AkQk+1Ak + CkC
k Qk = 0, k = 0, 1, , N 1. (8)
2
The existence of Qk = Qk 0 implies that the limit
limM
Mj=0
AkAk+1 Ak+j1Ck+jCk+jAk+j1 Ak+1Ak
exists as M approaches infinity. Further more, it implies that
lA = maxi
i(A) < 1,
hence the system (1) is stable.Given T with the state-space realization (1), the left-outer-
inner factorization is computed in (6), where Wk is an unitarymatrix yielding (7). Thus, we have
Cv,kCv,k +Av,kA
v,k = I. (9)
It is clear that (9) is exactly the same as (8) with Qk = I , fork = 0, 1, , N 1. Thus Qk = I , for k = 0, 1, , N 1,are solutions for (8), resulting in
lAv = maxi
i(Av) < 1,
where Av = Av,k Av,k+N1 is the lifted state matrixfor the inner part. From the stability of causal system inDefinition 1, we can conclude that the inner factor with staterealization {Av,k, Bv,k, Cv,k, Dv,k} is stable. This proves thatthe inner factor derived by the outer-inner factorization isstable. The stability of the inner factor derived by the inner-outer factorization in [9] can be checked similarly. And itcan be proved that the inner factor derived by the inner-outerfactorization is also stable.
The inverse of the inner factor is anti-causally realized andhas the state matrices {Av,k, Cv,k, Bv,k, Dv,k}. The existenceof Qk = Qk = I 0 also implies that
lAv = maxii(Av) < 1,
where Av = Av,k+N1 Av,k is the lifted state matrix for
the anti-causal inverse of the inner part. Hence, we can makethe following statement based on the stability of the anti-casualsystem defined in Definition 2.
Theorem 2: The anti-causal inverse system V = V of theinner factor V is stable. 2
The stability of V implies that the direct inverse V 1 ofthe inner factor V is unstable.
V. ERROR ANALYSIS
A. Zero-input and zero-state responses
The output of the LPTV system T is a combination of twoparts, the zero-input response and the zero-state response andcan be written as:
y = x0Tx + uT, (10)
where x0 is the unknown initial state value and Tx is thesystem zero-input operator, which can be expressed in termsof the state matrices as follows
Tx =[C0 A0C1 A0A1C2 A0A1 AL1CL
].
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For the system T being factorized into the outer and innerfactors, i.e. T = ToV , we consider the system outputs fromthe outer factor and from the inner factor, respectively, in thefollowing.
1). The output yo from the outer part
yo = xo,0To,x + uTo, (11)
where xo,0 is the unknown initial state value for the outer part,and To,x is the zero-input operator, which can be written as
To,x =[Co,0 Ao,0Co,1 Ao,0Ao,1 Ao,L1Co,L
].
And To is the outer factor of the transfer operator, which isgiven by
To,i,j ={
Do,i, i = j,Bo,iAo,i+1 Ao,j1Co,j , i < j.
2). The output yo from the outer part is the input to theinner part, resulting in the following system output
y = xv,0Tv,x + yoV, (12)
where xv,0 is the unknown initial state value for the inner part,and Tv,x is the zero-input operator of V represented by
Tv,x =[Cv,0 Av,0Cv,1 Av,0Av,1 Av,L1Cv,L
].
And V is the inner factor given by
Vi,j ={
Dv,i, i = j,Bv,iAv,i+1 Av,j1Cv,j , i < j.
In applying the inverse system with the anti-causal stablerealization to compute the system input u, the output signal yfrom the stable LPTV system T will pass through the anti-causal inverse of the inner factor first. It follows that theoutput of the anti-causal inverse inner factor is the input of thecasual inverse outer factor. In the following, we will discuss,respectively, errors raised in the anti-causal inverse of the innerfactor and in the causal inverse of the outer factor.
B. Error in the anti-causal stable inverse of the inner factor
The state matrices of the anti-causal inverse of theinner factor are written as {Av,k, Bv,k, Cv,k, Dv,k} ={Av,k, Cv,k, Bv,k, Dv,k}. The following equations can then beobtained following from (7):
Av,kAv,k + Cv,kBv,k = I, Av,kCv,k + Cv,kDv,k = 0,Bv,kAv,k +Dv,kBv,k = 0, Bv,kCv,k +Dv,kDv,k = I.
(13)The periodic Lyapunov equation with Qk = I can be
formed by the first equation in (13), which is shown as
Av,kQk+1Av,k +Cv,kBv,k Qk = 0, k = 0, 1, , N. (14)Recursively using (14), we can get
Qk = Cv,k+1Bv,k+1 +
i=k+2Av,k+1Av,k+2 Av,i1Cv,iBv,iAv,i1 Av,k+2Av,k+1= I
(15)
We now express the anti-causal inverse output sequence yo as
yo = xv,LTv,x + yV , (16)
where xv,L is an unknown initial state value of the anti-causalinverse of the inner part, entries of the zero input responseTv,x are
Tv,x =[Av,LAv,L1 Av,1Cv,0 Av,LCv,L1 Cv,L
],
and entries of V are
Vi,j ={
Dv,i, i = j,Bv,iAv,i1 Av,j+1Cv,j , i > j.
The error in yo is its difference from yo in (11) which isobtained by using (11) and (12) as follows.
eo = yo yo,= xv,LTv,x + xv,0Tv,xV + yo[V V I].
The inversion computation can yield V V = I for aninfinitely long dada length. But with a real finite lengthcomputation, V R(L+1)(L+1), then V V 6= I . With theexpressions of the entries of the matrices V and V , we canobtain entries of the matrix V V as follows:
For i = j:
[V V ]i,i = Dv,iDv,i +Bv,i[Cv,i+1Bv,i+1 +L
k=i+2Av,i+1Av,i+2 Av,k1Cv,kBv,kAv,k1 Av,i+2Av,i+1]Cv,i.
If L, (15) can be applied, leading to[V V ]i,i = Dv,iDv,i +Bv,iCv,i = I.
For i > j:
[V V ]i,j = Bv,i[Cv,i+1Bv,i+1 +L
k=i+2Av,i+1 Av,k1Cv,kBv,kAv,k1 Av,i+1 I]Av,i Av,j+1Cv,j .
If L, (15) can be applied, leading to[V V ]i,j = 0.
For i < j:
[V V ]i,j = Bv,iAv,i+1 Av,j [Cv,j+1Bv,j+1 +L
k=j+2Av,j+1 Av,k1Cv,kBv,kAv,k1 Av,j+1 I]Cv,j .
If L, (15) can be applied, leading to[V V ]i,j = 0.
For practically feasible computing with the anti-causal real-ization in finite L computation steps, V R(L+1)(L+1) andV V 6= I . For small values of i, j, [V V ]i,j is approaching zerosince lAv < 1. And for large values of i, j, [V V ]i,j becomesnon-negligible.
Even thought we force the initial state values of the innerpart and the anti-causal inverse of the inner part, xv,0 and xv,Lrespectively, to be zero, the system error is not equal to zero.The error is then propagated to the causal inverse of the outerfactor and can be amplified in this part.
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C. Error analysis for causal stable inverse of the outer factor
For the causal stable inverse of the outer factor, the statematrices are given as
Ao,k = Ao,k Co,kD1o,kBo,k, Bo,k = D1o,kBo,k,Co,k = Co,kD1o,k, Do,k = D1o,k,
yielding,
Ao,k + Co,kBo,k = Ao,k Co,kD1o,kBo,k + Co,kD1o,kBo,k= Ao,k
(17)The transfer operator To of the causal inverse of the outerfactor has the entries shown as:
To,i,j ={
Do,i, i = j,Bo,iAo,i+1 Ao,j1Co,j , i < j.
Hence, it can be shown that for the finite length computing,ToTo = I is satisfied. Entries of ToTo satisfy the followingresults.
For i = j:[ToTo]i,i = Do,iDo,i = I,
For i > j:[ToTo]i,j = 0,
For i < j:
[ToTo]i,j = Do,iBo,iAo,i+1 Ao,j1Co,j+BiCo,i+1Bo,i+1Ao,i+2 Ao,j1Co,j+Bo,iAo,i+1Co,i+2Bo,i+2Ao,i+3 Ao,j1Co,j...+Bo,iAo,i+1Ao,i+2 Ao,j1Co,jDo,j .
Recursively using (17) can result in [ToTo]i,j = 0, for i > j.The output from the inverse of the outer part is written as
u = xo,0To,x + (yo + eo)To,= xo,0To,x + xo,0To,xTo + uToTo + e1To,
(18)
where xo,0 is the unknown initial state value for the inverseof the outer part, and the zero input response To,x can beexpressed as
To,x =[Co,0 Ao,0Co,1 Ao,0Ao,1 Ao,L1Co,L
].
The system error e is the difference between the input signaland the reconstructed signal u, which can be presented asfollows.
e = u u,= xo,0To,x + xo,0To,xTo + u[ToTo I] + e1To,= xo,0To,x + xo,0To,xTo + u[ToTo I]
+xv,LTv,xTo + xv,0Tv,xV To + yo[V V I]To= xo,0To,x + xv,LTv,xTo + xv,0Tv,xV To
+xo,0To,xV V To + uTo[V V I]To.If the initial state values of the original and the inverse systemsare all zero, the system error e is simplified to
e = uTo[V V I]To. (19)
Thus, if we do not consider the system error generated bythe initial state values, the inverse of the outer part, whichis a causal realization, will not generate any additional errorexcept acting on the error eo from the inverse of the innerpart and passing it to the output. Because of the finite stepcomputation, V V I 6= 0, which generates inversion errorwithin the anti-causal inverse system of the inner factor.
D. Error reduction
In the error analysis of the anti-causal stable inverse of theinner factor part, we have pointed out that for small i, j values,[V V ]i,j is approaching zero since lAv < 1, and for large i, jvalues, [V V ]i,j become non-negligible. This shows that thesystem error is negligible when k is small and then becomelarger and larger with k being increased. In order to reducethe non-negligible system error, zeros can be appended to theinput sequence. Let the number of zeros appended to the inputsequence is K. To show that the inversion error is dependenton the number of zeros appended to the input sequence, weexamine the last column of V V . For i < (L+ 1),
[V V ]i,(L+1) = Bv,i1Av,i Av,L[Cv,L+1Bv,L+1+K
k=2Av,L+1 Av,L+k1Cv,L+kBv,L+kAv,L+k1 AL+1 I]Cv,L,
and
[V V ](L+1),(L+1) = Dv,LDv,L +Bv,L[Cv,L+1Bv,L+1+K
k=2Av,L+1 Av,L+k1Cv,L+kBv,L+kAv,L+k1 AL+1 I]Cv,L
For a sufficiently large K, it can be shown that
[Cv,L+1Bv,L+1 +K
k=2Av,L+1 Av,L+k1Cv,L+kBv,L+kAv,L+k1 AL+1] I,
[V V ]i,(L+1) 0 and [V V ](L+1),(L+1) I , hence the systemerror (19) is reduced.
This shows that by appending a sufficiently large numberof zeros in the input sequence will reduce the inversion errorto some sufficiently small value. Practically, number K canbe dependent on the tolerance level of the system error andthe total available computational time.
VI. EXAMPLEIn this section, Example 1 shows the mix-causal inverse
system error with zero initial state values. Example 2 is anextension of Example 1, showing the reduced system error byusing appended zeros in the input sequence approach.
Example 1. Consider a 2-periodic system with the follow-ing state matrices:
for k=0,2,4,...
Ak =[ 0.5 0.548
0 0.5], Ck =
[1
0.548
],
Bk =[1 0.548
], Dk = 1;
for k=1,3,5,...
Ak =[0.5 2.3330 0.333
], Ck =
[11
],
Bk =[2.5 2.333
], Dk = 1.
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The first step is to obtain the inner and outer factors of thegiven system using (6). The input signal u is the gaussianrandom signal with zero mean and variance of 1 with lengthof 1 500. The output y of the given 2-periodic system isobtained by (10). The output of the anti-causal inverse isgiven by (16) with y as the input. The reconstructed signalu is obtained by (18). All the initial state values are set tobe zero. The simulation is done using Matlab program. Theinput signal, the reconstructed signal and the system error areshown in Fig.1.
0 50 100 150 200 250 300 350 400 450 5004
2
0
2
4input signal
0 50 100 150 200 250 300 350 400 450 5004
2
0
2
4reconstructed signal
0 50 100 150 200 250 300 350 400 450 5002
1
0
1
2error
Fig. 1. Input-Output and mix-causal inverse system error
Example 2. This example is to show a result of appendingzeros to the input sequence. We append 20 zeros to the laststage of the input sequence of Example 1 and its effect can beseen by comparing the system errors in Fig. 1 and Fig. 2. InFig.2, it is shown that the error is translated to the extendedtime period corresponding to the time period of the appendedzeros. This practically reduces the inversion error within thetime interval of the effective input sequence.
VII. CONCLUSION
In this paper, we have analyzed the stability of the inverseof the LPTV system with an anti-causal realization. The outer-inner factorization technique is employed to derive the inversesystem. The stability of the outer factor and the inverse ofthe outer factor is analyzed and established using a periodicLyapunov equation technique. This is further applied to thestability analysis of the inner factor and its inverse, whichis realized anti-causally with the state matrices being theHermitian conjugate of that of the inner factor. Its stabilityis also shown using the Lyapunov equation technique.
We have also studied the error of the inverse system whichcontains the anti-causal inverse of the inner factor and thecausal inverse of the outer factor. The error of the inversesystem is derived in terms of the initial state values and thezero state response. Because of the anti-causal nature of theanti-casual inverse of the inner factor, although V V = I
0 100 200 300 400 500520 6004
2
0
2
4input signal
0 100 200 300 400 500520 6004
2
0
2
4reconstructed signal
0 100 200 300 400 500520 6002
1
0
1
2x 105 error
Fig. 2. Input-Output and mix-causal inverse system error with zerosappending
for infinitely long signal computing, V V in general is notequal to I for finite length computing of signals. Hence thesystem inversion error is inevitable by using the anti-causalrealization. It is also shown that the inversion error can bereduced by appending zeros to the input sequence.
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